Conformity and bounded rationality in games with many players

Conformity and Bounded Rationality

in Games with Many Players

Edward Cartwright

And

Myrna Wooders

No 687

WARWICK ECONOMIC RESEARCH PAPERS

Conformity and bounded rationality in games

with many players.

Edward Cartwright

Department of Economics

University of Warwick

Coventry CV4 7AL, UK

[email protected]

Myrna Wooders

Department of Economics

University of Warwick

Coventry CV4 7AL, UK

[email protected]

http://www.warwick.ac.uk/fac/soc/Economics/wooders/

October 20, 2003

Abstract

1 Conformity, bounded rationality and equilibrium

We suggest that in games with many players common elements of bounded rational behavior are the use of pure strategies and conformity in the sense that a player is inclined to choose the same strategy as players he perceives as similar to himself. If this is the case, and we take Nash equilibrium as an outcome of fully rational behavior, then the consistency of boundedly rational behavior and rationality requires the existence of an approximate Nash equilibrium exhibiting conformity. In this paper we provide a family of games with many players where the desired equilibrium exists.

An important question not addressed in CW is whether, for games with many players - large games - the parameters describing individual games, and thus", can be chosen to be small. In this paper we introduce a frame-work of games with incomplete information and demonstrate conditions un-der which the numbers of player classes can be chosen to be relatively small while the distance between any two players of the same class is small. Con-formity and existence of an approximate equilibrium in pure strategies then follows from our prior results.

To treat a family of games of incomplete information we take as given a set of attributes-, a set of player typesT and a set of actionsA:A player’s attribute is assumed to be publicly observable while a player’s type, deter-mined by nature, is not. A universal payo¤ functionhdetails the payo¤ of a player as a function of his attribute, type and action and the attributes, types and actions of the complementary player set. A universal beliefs func-tion b details the probability distribution over type pro…les - players are assumed to have consistent beliefs with respect to this distribution. We re-fer to the tupleG= (-;A;T; b; h) as a non-cooperative pregame. A player

set and an attribute function, assigning an attribute to each player, induce, through the pregame, a game.

is independent of the number of players. Thus, in large populations societies must also be large.

Following CW we permit an endogenous assignment of roles within a society. A player is assigned a role according to some probability distribution determined by the society and can make his action choice conditional on his role. This approach allows us to model the case where players could be seen as conforming (or belonging to the same society) even though they may perform di¤erent actions: Given that players can make action choice conditional on role or type, two players can play the same strategy and yet (because they have been allocated di¤erent roles or types) play di¤erent actions.

As well as treating the bounded rationality of conformity in pure gies we also treat in isolation the bounded rationality of playing pure strate-gies and the bounded rationality of conformity. In both cases we provide su¢cient conditions on a pregame for the existence of an approximate Nash equilibrium satisfying the desired properties - either one in pure strategies or one consistent with conformity.

Elaborating further on the prior literature, WCS provide a family of games with many players for which there exists an approximate Nash equi-librium in pure strategies that partitions the player set into a bounded num-ber of societies. Two limitations of the results due to WCS are: …rst, it only treats games of perfect information which, amongst other things, does not allow us to model an assignment of roles within a society. Second, the bound on number of societies is proportional to the number of strategies; given that the framework of WCS can be extended to allow a countable set of strategies (see Cartwright and Wooders 2003b) this appears a signi…cant limitation.

This is done by introducing the concept of a(±; Q)-class game - any …nite

game is (±; Q)-class game for some values of ± and Q. Given a(±; Q)-class game a bound on"permitting existence of a Bayesian Nash"-equilibrium in pure strategies consistent with social conformity, is provided as a function of ± and Q. The approach of CW has the advantage of treating individual games and permitting incomplete information. In addition, CW are able to bound the number of societies independently of the number of strate-gies. CW, however, only put a bound on the " for which there exists an "-equilibrium satisfying the desired properties - they do not provide condi-tions under which the"is small.

In this paper we address some of the issues that arise from WCS and CW. First, we extend the pregame framework of WCS to permit incomplete information. We then demonstrate a connection between(±; Q)-class games and games induced from a pregame. This allows us to apply the results of CW and in doing so provide a family of games where the use of pure strate-gies and conformity can be consistent with individually rational behavior. Further, in the results of this paper, the number of societies is bounded independently of the number of strategies.

We proceed as follows: Section 2 introduces de…nitions and notation and Section 3 reviews the de…nition of a(±; Q)-class game. In Section 4 we treat conformity, in Section 5 we treat pure strategies and in Section 6 we treat conformity in pure strategies. In Section 7 we conclude.

2 De…nitions and notation

with the de…nition of a Nash equilibrium.

2.1 A Bayesian Game

A Bayesian game ¡ is given by a tuple (N; A; T; g; u) where N is a …nite

player set, A is a set of action pro…les, T is a set of type pro…les, g is a

probability distribution over type pro…les and u is a set of utility functions. We de…ne these components in turn.

LetN =f1; :::; ngbe a …nite player set, letAdenote a …nite set ofactions

and let T denote a …nite set oftypes. ‘Nature’ assigns each player a type. Informed of his own type but not the types of his opponents, each player chooses an action. We say that a game isa game of perfect information if

jT j = 1. Let A ´ AN _{be the set of action pro…les and let T ´ T}N _{be the} set oftype pro…les. Given action pro…le aand type pro…letwe letai and ti denote respectively the action and type of playeri2N.

A player’s payo¤ depends on the attributes, actions and types of players. Formally, in game ¡, for each player i 2 N there exists a utility function

ui :A£T !R. In interpretation ui(a; t) denotes the payo¤ of player i if the action pro…le isaand the type pro…le t. Letu denote the set of utility functions.

A player, once informed of his own type, selects an action without know-ing the types of the other players. A player therefore forms beliefs over the types he expects others to be. These beliefs are represented by a functionpi wherepi(t_¡ijti) denotes the probability that playeriassigns to type pro…le

(ti; t¡i) given that i is of type ti. Throughout we will assume consistent

beliefs. Formally, for some probability distribution g over type pro…les, we assume:

pi(t¡ijti) = P g(ti; t¡i) t0_¡i2T_¡ig(ti; t

0 ¡i)

for all i2N and ti 2 T.1

2.2 Pregames

Let- be a compact metric space, called anattribute space and letN be a …niteplayer set. A function ®mapping from N to- is called an attribute function. The pair (N; ®) is a population. In interpretation, an attribute function ascribes an attribute to each player in a population. Taking as given a …nite set of actions A and types T a population (N; ®) induces a Bayesian game¡(N; ®)´(N; A; T; g®; u®) as we now formalize.

Denote by W the set of all mappings from -£ A £ T into Z+; the

non-negative integers. A member of W is called a weight function. Given population(N; ®) we say that a weight function w®;a;t is relative to action

pro…le a and type pro…le t if and only if:

w®;a;t(!; al; tz) =¯¯¯ni2N :®(i) =!; ai=al and ti =tzo¯¯¯:

Thus,w(!; al_{; t}z_{) denotes the number of players with attribute ! and type} tz who take actional.

A universal payo¤ function h maps -£ A £ T £W into R+, the

non-negative real numbers. The function hwill determine payo¤ functions for every game induced by the pregame. Given a population(N; ®), the payo¤ of a player will depend on his attribute, his action, his type and the weight function induced by the attributes, actions and types of the complementary player set. Formally:

u®i(a; t)

def

= h(®(i); ai; ti; w®;a;t):

Denote by D the set of all mappings from -£ T into _Z+. A member

of D is called a type function. Given population (N; ®) we say that type 1_{We assume that the denominator of (1) is always positive - i.e. there is positive}

functiond®;t isrelative to type pro…le t if:

d®;t(!; tz) =jfi2N :®(i) =! and ti =tzgj:

Thus, d®;t(!; tz_{) denotes the number of players with attribute ! and type} tz_.2 _{A universal beliefs function b maps D into [0;1]. The value b(d®;t) is}

interpreted as the probability of type pro…let. Formally:

g®(t)def= b(d®;t)

whereg® is the probability distribution over type pro…les induced byband ®. Players are assumed to have consistent beliefs with respect to g®. It is important to realize the di¤erences between functions g® _{and b. Function} g® is de…ned relative to a population(N; ®)and its domain isTN_{. Function} b, however, is de…ned independently of any speci…c game and has domain

D.

A pregame is given by a tuple

G= (-;A;T; b; h);

consisting of a compact metric space-of attributes, …nite action and type

sets A and T, a universal beliefs function b :D ¡![0;1] and a universal payo¤ functionh: -£ A £ T £W ¡!R+. As discussed above we refer to

a population(N; ®) as inducing, through the pregame, a Bayesian game

¡(N; ®)´(N; A; T; g®; u®):

2.3 Strategies and expected payo¤s

Take as given a population(N; ®)and induced Bayesian game(N; A; T; g®_{; u}®_). As discussed above, knowing his own type, but not those of his opponents a player chooses an action. A pure strategy details the action a player will

2_{Note thatd}

take for each type tz 2 T and is given by a function sk : T ! A where sk_(tz_{) is the action taken by the player if he is of type t}z_{. Denote the set} of pure strategies byS and let K=jAjjT j=jSj denote the number of pure strategies.

A (mixed) strategy is given by a probability distribution over the set of pure strategies. The set of strategies is denoted by¢(S). Given a strategyx we denote byx(sk_{)the probability that a player takes pure strategy s}k _{2 S.} We denote by x(aljtz) the probability that a player takes action al given that he is of typetz_{. Let§ = ¢(S)}N _{denote the set ofstrategy vectors. We} refer to a strategy vector masdegenerate if for all i2N and tz 2 T there exists some actional _{for which mi(a}l_jtz_{) = 1.}

We assume that players are motivated by expected payo¤s. Given a strategy vector ¾, a type tz _{2 T and beliefs about the type pro…le p}®

i the probability that player i puts on the action pro…le-type pro…le pair a = (a1; :::; an) and t= (t1; :::; ti¡1; tz; ti+i; :::; tn)is given by:

Pr(a; t¡ijtz)def= p®i(t¡ijtz)¾1(a1jt1):::¾i(aijtz):::¾n(anjtn).

Thus, given any strategy vector ¾, for any type tz 2 T and any player i of type tz_{, the expected payo¤ of player ican be calculated. Let U}®

i (¢jtz) :

§ !R denote the expected utility function of player i conditional on his type beingtz _where:

Ui®(¾jtz)

def

= X

a2A

X

t_¡i2T¡i

Pr(a; t¡ijtz)u®i(a; tz; t¡i).

Denote by EW the set of functions mapping -£ A £ T into R+, the

non-negative reals. We refer to ew; eg 2 EW as expected weight functions. Given a population(N; ®)we say that an expected weight functionew®;¾ is relative to strategy pro…le¾ if and only if:

ew®;¾(!; al; tz) =X

a2A

X

t2T

for all !; al and tz. Thus, ew®;¾(!; al; tz) denotes the expected number of

players of attribute! who will have type tz _{and play action a}l_{. Note that} this expectation is taken before any player is aware of his type.

2.4 Nash equilibrium

The standard de…nition of a Bayesian Nash equilibrium applies. A strategy vector ¾ is a Bayesian Nash "-equilibrium (or informally an approximate Bayesian Nash equilibrium) if and only if:

U_i®(¾i; ¾_¡ijtz)¸Ui®(x; ¾¡ijtz)¡"

for allx2¢(S), alltz _{2 T and for alli2N. We say that a Bayesian Nash} "equilibrium mis aBayesian Nash "-equilibrium in pure strategies ifmis degenerate.

3

(

±; Q

)

-class games

Informally, a game ¡(N; ®) = (N; A; T; g®; u®) is a (±; Q)-class game if the

population N can be partitioned into Q subsets N1; :::; NQ; called classes,

where (1) any two players in the same class are ‘±-substitutes’ for each other and (2) roughly, the payo¤ to a player depends only on his own strategy choice and the ‘aggregate strategy’ of the players in each class. The concept of a(±; Q)-class game was introduced in CW.

To formally de…ne a(±; Q)-class game we require notions of approximate

substitute players. Take as given a game ¡(N; ®) and a partition of the player setN =fN1; :::; NQg.

Partition N is a ±I-interaction substitute partition when: For any two strategy vectors¾1_{; ¾}2_2§if:

X

i2Nq

¾1i(sk) =

X

i2Nq

for allNq and allsk2 S, then:

¯

¯U_i®(x; ¾1_¡ijtz)¡U_i®(x; ¾2_¡ijtz)¯¯·±I

for any playeri2N and any strategyx2¢(S).

Informally, N is a ±I-interaction substitute partition if a player’s payo¤ changes by at most±I when other players of the same class ‘exchange’ strate-gies, with his own strategy choice held constant.

Partition N is a ±P-individual substitute partition when: For any Nq, for any two players i; j2Nq and for any strategy vector¾2§such that

¾i=¾j:

¯ ¯_U®

i (x; ¾¡ijtz)¡Uj®(x; ¾¡jjtz)

¯ ¯_·±P

for any strategyx2¢(S).

Informally,N is a±P-individual substitutepartition if the payo¤s of any two players in the same class, when they both play the same strategy and the strategies of other players are held constant, are within±P.

PartitionN is a±C-strategy switching partition when: For any two strategy vectors¾1; ¾2 2§ if:

X

i2Nq

¯ ¯

¯¾1i(sk)¡¾2i(sk)

¯ ¯

¯·1, (2)

for allNq and allsk2 S then:

¯ ¯_U®

i (x; ¾1¡ijtz)¡Ui®(x; ¾2¡ijtz)

¯ ¯_·±C

Thus, given a smallproportional change in the ‘aggregate strategy’ of a class, ifN is a±C-strategy switching partition then payo¤s will change by at most ±C.

Game ¡(N; ®) is said to be a(±I; ±P; ±C; Q)-class game if there exists a partitionN of the player set into Q classes such that N is a ±I-interaction substitute partition, a ±P-individual substitute and a ±C-strategy switching partition. If±I; ±P; ±C ·± then we refer to ¡(N; ®) as a (±; Q)-class game. Given a (±; Q)-class game ¡(N; ®) we refer to a partition N as a proper partition of the player set into classes if it is a ±-substitute partition and ±-strategy switching partition.

4 Games With Many Players

We will assume throughout a relatively mild continuity property with respect to attributes. This assumption, introduced in WCS, dictates that a player’s payo¤ is relatively invariant to a small perturbation of the attributes of players (including himself). Such an assumption would be satis…ed, for example, in a private goods economy where individual preferences depend only on own consumption of commodities. Formally:

Continuity in attributes: Pregame G = (-;A;T; b; h) is said to satisfy

continuity in attributes when: for any " > 0 and any two induced games ¡(N; ®) and ¡(N; ®), if, for alli2N,

dist(®(i); ®(i))< "

then, for alli2N, all tz 2 T and any strategy vector¾ 2§:

¯ ¯_U®

i (¾jtz)¡Ui®(¾jtz)

¯ ¯_{< ":}

change in attributes while the strategies are held constant. The assump-tion comprises essentially two distinct elements: (1) A player should be relatively indi¤erent to small changes in the attributes of others - this would suggest, amongst other things, that the probability distribution over types is largely una¤ected by a small change in the attribute function. (2) Players with similar attributes receive similar payo¤s. This later point will clearly have a role to play in demonstrating the existence of±P-individual substitute partitions for small values of ±.

4.1 Societies

We de…ne a society. Given a game (N; ®) and a strategy vector ¾ we in-terpret a set of players D as a society if (i) there exists some strategy x 2 ¢(S) such that ¾i = x for all i 2 D, and (ii) for any player i 2 N, if ®(i) 2 con(®(D)) then i 2 D.3 _{Thus, any two players belonging to a}

society D must play the same strategy. Furthermore, to any societyD we can associate a convex subset -D of attribute space -with the properties that any player belonging toD has an attribute in-D while there exists no player who has an attribute in-D that does not belong to D. 4

We say that a strategy vector ¾ induces a partition of the player set intoQ societies if there exists a Qmember partition of the player setN =

fN1; :::; NQg such that each subsetNq is a society.

Given a population(N; ®) we say that a partitionN =fN1; :::; NQgis a partition of (N; ®)into convex subsets if there exists a partitionf-1; :::;-Qg of-into convex subsets with the property that ifi2Nq then®(i)2-q for

3_{Wherecon(®(D))denotes the convex hull of®(D).}

4_{This is a stronger notion of conformity than used by WCS. In WCS condition (ii)}

all i2N.5

4.2 Conformity

In this section we demonstrate that for a large family of games we can put a boundQ on the number of societies, whereQ is independent of population size, such that any game within this family has an approximate Nash equi-librium that partitions the population into at most Q societies. Note that in this section we make no assumption that players use pure strategies.

We introduce a second assumption:

Risk Neutrality property: We say that a pregame G satis…es the risk neutrality property when: for any population (N; ®) and any two strategy pro…les ¾; ¾2§N _{with expected weight functions ew®;¾ and ew®;¾} respec-tively, where:

ew®;¾(!; al; tz) =ew®;¾(!; al; tz)

for all !; al and tz, if ¾i =¾i then:

Ui®(¾jtz) =Ui®(¾jtz) for anytz_{2 T.}

The risk neutrality property requires players to be risk neutral with respect to the strategies of others. For example, consider two players iand j who both have attribute! and consider some other playerl. The risk neutrality property dictates that playerlshould be indi¤erent as to whether (i) player iplays strategys1 and playerjplays strategys2, (ii) playerj plays strategy s1 and player i plays strategy s2, and (iii) both players choose strategy

5_{Of course the sets}

-q are required to be onlyrelatively convex since -may not be

s1 and s2 with probability one half. There are many instances where this assumption would appear mild - we consider one case later.

Before stating our …rst Theorem, we recall that it follows from Theorem 2 of CW that any (0;0; ±; Q(¿))-class Bayesian game has a Bayesian Nash equilibrium (a0-equilibrium) with the property that any two players in the same class play the same strategy. Our …rst Theorem demonstrates that for any game induced by a pregame there is a ‘near-by’(0;0; ±; Q) game for some± and Q. The existence of such games allows us to infer properties of games induced by pregames.

Theorem 1: Consider a pregameG = (-;A;T; b; h) that satis…es the risk

neutrality property. Given real number ¿ > 0 there is a real number Q(¿)

such that for any population(N; ®)there is another population (N; ®)

sat-isfyingdist(®(i); ®(i))< ¿ for alli2N and, for some±, the induced game

¡(N; ®) is a (0;0; ±; Q(¿))-class Bayesian game. Furthermore, there exists a partition N of N that is both a proper partition into classes for game

¡(N; ®) and a partition of(N; ®) into convex subsets.

Proof: Partition-into convex subsets-1; :::;-Qeach of diameter less than ¿ >0. For each subset-qchoose and …x an attribute!q 2-q. Consider an arbitrary game¡(N; ®). De…ne attribute function®as follows for alli2N:

®(i) =!q if and only if®(i)2-q.

Clearlydist(®(i); ®(i))< ¿ for alli2N. For each q, de…ne Nq=fi2N :

®(i) =!qg. We conjecture that the partitionN ´ fN1; :::; NQgsatis…es the

¡(N; ®). Thus, game ¡(N; ®) is a (0;0; ±; Q)-class Bayesian game and N

is a proper partition of N. It is immediate that N partitions (N; ®) into convex subsets.¥

Theorem 1 leads to Proposition 1, a consequence of the above result and Theorem 2 of CW:

Proposition 1: Consider a pregame G = (-;A;T; b; h) that satis…es the

risk neutrality property and continuity in attributes. Given real number " > 0 there exists real number Q1(") > 0 such that for any population (N; ®) the induced game ¡(N; ®) has a Bayesian Nash "-equilibrium that induces a partition of the player set intoQ1(") societies.

Proof: Given an " > 0, de…ne ¿ ´ 1

2". By Theorem 1 there exists real

number Q(¿) such that for any population (N; ®) there exists a popula-tion (N; ®) such that maxi2Nfdist(®(i); ®(i))g < ¿ and the induced game

¡(N; ®) is a(0;0; ±; Q)-substitute Bayesian game for some ±. Further there exists a proper partition of N into classes N for game ¡(N; ®) that is a

partition of (N; ®) into convex subsets. Theorem 2 of CW states that any

(0;0; ±; Q)-class Bayesian game has a Bayesian Nash 0-equilibrium m with the property that any two players of the same class play the same strategy. Thus, game¡(N; ®)has a Bayesian Nash0-equilibriummwith the property that any two players of the same class play the same strategy. By continuity in attributes, for alli2N:

¯

¯U_i®(x; m_¡ijtz)¡Ui®(x; m¡ijtz)

¯ ¯< "

2:

for all x2¢(S) and tz2 T. Thus:

If the number of attributes is …nite then we can go further. Of course with a …nite number of attributes conformity is less interesting. The following Theorem, also a consequence of Theorem 2 of CW, simply states that there exists a possibly mixed strategy where all players who are identical play the same strategy.

Proposition 2: Consider a pregame G that satis…es the risk neutrality property and where the number of attributes is a …nite integer Q. For any population (N; ®) the induced game ¡(N; ®) has a Bayesian Nash 0 -equilibrium that induces a partition of the player set intoQ societies.

Proof: Take as given game ¡(N; ®). Let- =f!1; ::::; !Qg be the space of

attributes and let N =fN1; :::; NQg denote the partition of the player set

wherei2Nq if and only if®(i) =!q. PartitionN is a 0-interaction substi-tute partition and a 0-substitute partition. Thus, ¡(N; ®) is a(0;0; ±; Q)

-class Bayesian game. Theorem 2 of CW states that any (0;0; ±; Q)-class Bayesian game has a Bayesian Nash0-equilibriummwith the property that any two players of the same class play the same strategy. This completes the proof.¥

5 Pure Strategy Equilibrium

The risk neutrality property proves insu¢cient for the existence of an ap-proximate Nash equilibrium in pure strategies. We therefore introduce a stronger large game property. First, taking a population(N; ®)as given, we

de…ne a metric on the spaceEW® of expected weight functions:

dist2(ew; eg) = 1

jNj

X

!2®(N)

X

al_2A

X

tz_2T

¯ ¯

¯ew(!; al; tz)¡eg(!; al; tz)

¯ ¯ ¯

are playing each action are close.

Large game property: We say that a pregame G satis…es the large game property when: for any" >0, any population (N; ®) and any two strategy pro…les¾; ¾2§N where:

dist2(ew®;¾; ew®;¾)< "

if¾i=¾i then:

jUi®(¾jtz)¡Ui®(¾jtz)j< ": for anytz2 T.

If a pregame satis…es the large game property then we can think of payo¤ functions as satisfying one principal condition - a player is nearly indi¤erent to a change in the proportion of players of each attribute and of each type playing each action (provided his own strategy is unchanged); thus, the behavior of no one individual or small group of individuals can have large e¤ects on a player’s payo¤. This contrasts with the risk neutrality property where one individual can have a large in‡uence.6 _{Risk neutrality is also}

required to hold under the large game property but an assumption of risk neutrality is mild in this context; with large player sets, …nite sets of pure strategies and …nite types the law of large numbers dictates that the actual proportion of players playing each action will, with high probability, be close to the expected proportion (Kalai 2002).

We state our second theorem:

Theorem 2: Consider a pregame G that satis…es the large game property. Given real numbers ± > 0 and ¿ > 0 there are integers ´(±; ¿) and Q(±; ¿)

such that for any population (N; ®), where jNj > ´(±; ¿), there exists a

similar population (N; ®) with dist(®(i); ®(i)) < ¿ for all i 2 N and the induced game ¡(N; ®) is a (±; Q(±; ¿))-substitute Bayesian game. Further there exists a proper partition into classes N for game ¡(N; ®) that is a

partition of(N; ®) into convex subsets.

Proof: Suppose that the statement of the lemma is false. Then there is some ± > 0 and ¿ > 0, such that for any real number Q and for each integerº there is a population(Nº_{; ®}º_{) where jN}º_{j > º and such that for} no population (Nº; ®º) where maxi2Nfdist(®(i); ®(i))g< ¿ is the induced game¡(Nº_{; ®}º_{) a (±; Q)-substitute Bayesian game.}

Partition -into convex subsets -1; :::;-Q each of diameter less than¿. To each subset-q choose and …x an attribute !q. For each(Nº; ®º) de…ne the attribute function®º as follows: for alli2Nº:

®º(i) =!q if and only if®(i)2-q. Given game (Nº_{; ®}º_{) let N}º _{= fN}º

1; :::; NQºg denote the partition of the player set such that i 2 Nqº if and only if ®º(i) = !q. We note that the value Q is …xed independently of the game (Nº_{; ®}º_{). The partition N}º is a 0-individual substitute partition for all º and, given the large game property, a±-interaction substitute partition. Also, for su¢ciently large º, by the large game property, Nº _{is a±-strategy switching partition. Thus,} game¡(Nº_{; ®}º_{) is a(±; Q)-substitute Bayesian game.¥}

Our third proposition demonstrates the existence of an approximate Nash equilibrium in pure strategies and obtains a puri…cation result as a consequence of the puri…cation result in CW for(±; Q) class-games.

number ´2(") >0 such that any induced game ¡(N; ®) wherejNj> ´2(")

has a Bayesian Nash"-equilibrium in pure strategies.

Proof: Let± ´ 16". By Theorem 2 there are real numbers´andQsuch that

for any population(N; ®), where jNj> ´, there exists a population (N; ®)

such that maxi2Nfdist(®(i); ®(i))g< ± and the induced game ¡(N; ®) is a

(±; Q)-substitute Bayesian game. Theorem 1 of CW states that any(±; Q)

-class game has a Nash4±-equilibrium. Letm be a Nash 4±-equilibrium of game¡(N; ®). By continuity in attributes, for all i2N:

¯ ¯_U®

i (x; m¡ijtz)¡Ui®(x; m¡ijtz)

¯ ¯_{< ±:}

for all x2¢(S) and tz2 T. Thus:

U_i®(mjtz)> U_i®(x; m_¡ijtz)¡6± for all x2¢(S). This completes the proof.¥

6 Conformity in Pure Strategies with Roles

Following the approach of CW we consider the possibility that players may conform in their choice of strategy yet play di¤erent actions. The existence of imperfect information permits this as a player’s action is conditional on his type. We assume that players can endogenously create imperfect in-formation through an allocation of roles within a society. To simplify the analysis we assume that play ‘begins’ with a game of perfect information.

Take as given a pregame G = (-;A;T; b; h) where jT j = 1. Games induced through this pregame are games of perfect information. Assume that there exists a set of rolesR=fr1; :::; rKg. Consider game¡(N; ®). Let

pro…ler. We consider a Bayesian game with endogenous roles ¡(f)(N; ®).

In game ¡(f)(N; ®) roles are (Harsanyi) types. Thus, roles are randomly allocated to players, a player can make his action choice conditional on his role and makes his choice of action knowing his role but not those of players in the complementary player set. A player’s payo¤, however, does not depend directly on the role pro…le. We assume that players have consistent beliefs with respect to the distribution over rolesf. Formally, we can de…ne game¡(f)(N; ®) = (N; A; T(f); g®(f); u®(f))to satisfy:

1. T(f)´R, 2. for allr 2R,

g®(f)(r)´f(r)

3. u®_i(f)(a; r)´u®_i(a) for all a2A,r 2Rand all i2N.

Condition 1 states that roles are equivalent to types. Condition 2 states that players have consistent beliefs with respect to the distribution of roles. Condition 3 states that payo¤s are not directly e¤ected by the role pro…le.

We highlight that roles and a probability distribution over role pro…les are de…ned relative to a speci…c game¡(N; ®) rather than a pregame. This re‡ects the idea that roles are endogenously created within a population. Thus, it is more natural to think of a probability distribution over roles taking as given a speci…c game¡(N; ®). Note that this observation is also

re‡ected in the statement of Proposition 4, to follow.

are playing the same strategy are not exhibiting the same behaviour. This motivates our …rst condition.

Within class anonymity: A probability distribution over rolesf satis…es

within class anonymity if the probability that a player from a class Nq will have role rk _{is (a priori) identical for all players belonging to that class.} Formally, ifi; j2Nq for someq then:

X

r2R:ri=rk

f(r) = X

r2R:rj=rk f(r)

for all rk2 R.

To motivate the next requirement, consider the example of a male-female household following the roles of ‘he goes out to work, she stays home’. For this norm to be successful, it is necessary that no player, knowing the struc-ture of society – the number of players with each role in his or her class – after roles are assigned, wishes to change role assignment. This motivates a second condition.

Within class determination: Given a role pro…le r let z(r; k; q) be the number of players in classNq who have rolerk. A probability distribution over rolesf is within class determined if for any class Nq and for any two role pro…lesrandr, iff(r); f(r)>0thenz(r; k; q) =z(r; k; q)for all classes q and for all rk _{2 R.}

in the same class who play the same strategy are conforming to some norm of behavior.

Before stating our …nal result we introduce one further de…nition. We use the concept of ex-post Nash equilibrium as introduced by Kalai (2002). Ex-post Nash implies that, knowing the action pro…le and the type pro…le, no player has a strong incentive to change her own action. Formally, given population ¡(N; ®) an action pro…le, type pro…le pair a; t is said to be "

ex-Post Nash if for all i2N:

u®i(a; t)¸ui®(al; a¡i; t)¡"

for all al 2 A. A strategy pro…le¾ is said to be aBayesian "ex-Post Nash

equilibriumif it yields an"ex-Post Nash action pro…le, type pro…le pair with probability one. If a strategy vector is a Bayesian ex-Post Nash equilibrium then, as discussed further by Kalai (2002), no player would wish to change his action after knowing the types (or roles) and the actions of the other players. The proof is based on that of CW (Theorem 3).

Proposition 4: Consider a pregame G that satis…es the large game prop-erty and continuity in attributes and where jT j= 1. Given a real number

" >0 there are real numbers´4(")>0 and Q4(") such that for any

popu-lation(N; ®) wherejNj> ´4(") there exists a probability distribution over

role pro…les f (that is within class anonymous and determined) with the property that game¡(f)(N; ®) has a Bayesian "ex-Post Nash equilibrium in pure strategies that induces a partition of the player set into at most Q4(") societies.

Proof: Let ± ´ 1

12". By Theorem 2 there are real numbers ´ and Q such

that for any population (N; ®), where jNj > ´, there exists a population

¡(N; ®) is a(±; Q)-substitute Bayesian game. Further there exists a proper

partition into classesN for game ¡(N; ®) that is a partition of (N; ®) into convex subsets.

Theorem 3 of CW states that any given any (±; Q)-class game ¡(N; ®)

and proper partitionN there exists a probability distribution over role pro-…lesf (that is within class anonymous and determined) such that¡(f)(N; ®)

has a Bayesian 10± ex–Post Nash equilibrium with the property that every player of the same class plays the same pure strategy. Let m be such an equilibrium of game¡(N; ®). By continuity in attributes, for alli2N:

¯

¯U_i®(x; m_¡ijtz)¡Ui®(x; m¡ijtz)

¯ ¯< ±:

for all x2¢(S) and tz2 T. Thus:

Ui®(mi; m¡ijtz)> Ui®(x; m¡ijtz)¡12± for all x2¢(S). This completes the proof.¥

7 Conclusion

In this paper we provide a family of games with many players for which there exists an approximate Nash equilibrium in pure strategies exhibiting conformity. A strategy vector exhibits conformity when the population could be partitioned into a relatively small number of societies - players in the same society play the same strategy and have similar attributes. The existence of roles within a society was permitted, thus allowing the possibility that players play the same strategy and yet perform di¤erent actions.

paper, however, extends that of WCS in considering games of imperfect in-formation. This allows a di¤erent interpretation of conformity and of a society. As a consequence we are able to bound the number of societies independently of the number of strategies (in contrast to WCS). In CW we treat individual games and provide a bound on the", depending on the parameters describing the game, allowing existence of a Nash"-equilibrium in pure strategies exhibiting conformity. CW do not, however, demonstrate that in large games this bound, and thus", can be taken to be small. This paper applies the results of CW in focussing on large games.

References

[1] Cartwright, E.J. and M.H. Wooders (2003a) “Bounded rational-ity in arbitrary games with many players,” Universrational-ity of War-wick Department of Economics Working Paper no 672. On-line at http://www2.warwick.ac.uk/fac/soc/economics/research/papers/twerp672-revised.pdf

[2] Cartwright, E.J. and M.H. Wooders (2003b) “On approximate equilib-rium in pure strategies in games with many players,” University of War-wick Working Paper (forthcoming).

[3] Kalai, E. (2002) “Ex-post stability in large games,” Northwestern Uni-versity Discussion Paper 1351.